A unified turbulence model bridging low and high Reynolds numbers: Integrating shell models with two-scale direct interaction approximation

Turbulence modeling across a wide range of Reynolds numbers remains a significant challenge in computational fluid dynamics. This study presents a unified turbulence model that integrates the GOY shell model with the Two-Scale Direct Interaction Approximation (TSDIA) to consistently capture both the inertial and dissipation ranges of the energy spectrum, including its temporal decay characteristics. Based on this theoretical formulation, a new eddy viscosity model is derived, bridging low- and high-Reynolds-number regimes while retaining physical consistency. The proposed model is validated through multiple benchmark cases. In backward-facing step flow, it significantly improves reattachment point prediction and turbulence statistics compared to the standard and RNG K-ε models. For flow over steep terrain, the model captures separation and reattachment behavior with higher fidelity, outperforming the conventional K-ε model. Furthermore, large eddy simulation (LES) of turbulent channel flow demonstrates that the proposed model reproduces mean flow profiles, turbulence intensities, and wall skin friction with accuracy comparable to DNS and the Smagorinsky model. It also enables consistent evaluation of the energy spectrum and transfer function across wide wavenumber ranges. These results confirm that the proposed framework not only strengthens theoretical consistency but also enhances numerical accuracy and robustness in practical simulations. By explicitly incorporating the energy cascade structure into turbulence modeling, the proposed approach offers a viable path toward unified, scalable, and physically grounded turbulence models for complex engineering and geophysical flows.